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Theorem uniqsALTV 37502
Description: The union of a quotient set, like uniqs 8774 but with a weaker antecedent: only the restricion of 𝑅 by 𝐴 needs to be a set, not 𝑅 itself, see e.g. cnvepima 37510. (Contributed by Peter Mazsa, 20-Jun-2019.)
Assertion
Ref Expression
uniqsALTV ((𝑅𝐴) ∈ 𝑉 (𝐴 / 𝑅) = (𝑅𝐴))

Proof of Theorem uniqsALTV
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecex2 37501 . . . . 5 ((𝑅𝐴) ∈ 𝑉 → (𝑥𝐴 → [𝑥]𝑅 ∈ V))
21ralrimiv 3144 . . . 4 ((𝑅𝐴) ∈ 𝑉 → ∀𝑥𝐴 [𝑥]𝑅 ∈ V)
3 dfiun2g 5033 . . . 4 (∀𝑥𝐴 [𝑥]𝑅 ∈ V → 𝑥𝐴 [𝑥]𝑅 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅})
42, 3syl 17 . . 3 ((𝑅𝐴) ∈ 𝑉 𝑥𝐴 [𝑥]𝑅 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅})
54eqcomd 2737 . 2 ((𝑅𝐴) ∈ 𝑉 {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅} = 𝑥𝐴 [𝑥]𝑅)
6 df-qs 8712 . . 3 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
76unieqi 4921 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
8 df-ec 8708 . . . . 5 [𝑥]𝑅 = (𝑅 “ {𝑥})
98a1i 11 . . . 4 (𝑥𝐴 → [𝑥]𝑅 = (𝑅 “ {𝑥}))
109iuneq2i 5018 . . 3 𝑥𝐴 [𝑥]𝑅 = 𝑥𝐴 (𝑅 “ {𝑥})
11 imaiun 7247 . . 3 (𝑅 𝑥𝐴 {𝑥}) = 𝑥𝐴 (𝑅 “ {𝑥})
12 iunid 5063 . . . 4 𝑥𝐴 {𝑥} = 𝐴
1312imaeq2i 6057 . . 3 (𝑅 𝑥𝐴 {𝑥}) = (𝑅𝐴)
1410, 11, 133eqtr2ri 2766 . 2 (𝑅𝐴) = 𝑥𝐴 [𝑥]𝑅
155, 7, 143eqtr4g 2796 1 ((𝑅𝐴) ∈ 𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  {cab 2708  wral 3060  wrex 3069  Vcvv 3473  {csn 4628   cuni 4908   ciun 4997  cres 5678  cima 5679  [cec 8704   / cqs 8705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8708  df-qs 8712
This theorem is referenced by:  imaexALTV  37503  rnresequniqs  37505  cnvepima  37510
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